Dynamics of Entanglement between a Quantum Dot Spin Qubit...

Hindawi Publishing CorporationAdvances in Mathematical PhysicsVolume 2010, Article ID 342915, 31 pagesdoi:10.1155/2010/342915

Research ArticleDynamics of Entanglement betweena Quantum Dot Spin Qubit and a Photon Qubitinside a Semiconductor High-Q Nanocavity

Hubert Pascal Seigneur,1 Gabriel Gonzalez,2, 3

Michael Niklaus Leuenberger,2, 3 andWinston Vaughan Schoenfeld1

1 CREOL College of Optics and Photonics, University of Central Florida, Orlando, FL 32826, USA2 NanoScience Technology Center, University of Central Florida, Orlando, FL 32826, USA3 Department of Physics, University of Central Florida, P. O. Box 162385, Orlando, FL 32816, USA

Correspondence should be addressed to Hubert Pascal Seigneur, [email protected]

Received 15 September 2009; Accepted 24 November 2009

Academic Editor: Shao-Ming Fei

Copyright q 2010 Hubert Pascal Seigneur et al. This is an open access article distributed underthe Creative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

We investigate in this paper the dynamics of entanglement between a QD spin qubit and a singlephoton qubit inside a quantum network node, as well as its robustness against various decoherenceprocesses. First, the entanglement dynamics is considered without decoherence. In the smalldetuning regime (Δ � 78μeV), there are three different conditions for maximum entanglement,which occur after 71, 93, and 116 picoseconds of interaction time. In the large detuning regime(Δ � 1.5 meV), there is only one peak for maximum entanglement occurring at 625 picoseconds.Second, the entanglement dynamics is considered with decoherence by including the effects ofspin-nucleus and hole-nucleus hyperfine interactions. In the small detuning regime, a decentamount of entanglement (35% entanglement) can only be obtained within 200 picoseconds ofinteraction. Afterward, all entanglement is lost. In the large detuning regime, a smaller amountof entanglement is realized, namely, 25%. And, it lasts only within the first 300 picoseconds.

1. Introduction

In order to continue satisfying Moore’s law and sustain technological growth, a quantumapproach that takes advantage of the wave nature of particles needs to be considered.Currently, the main problem is no longer the physical realization of the qubit, but ratherthe engineering of a practical quantum computing architecture or quantum network. As aconsequence of decisive factors such as on-chip implementation, efficient transfer of quantuminformation, scalability, CMOS compatibility, and cost, the general consensus is that variousimplementations of the qubit should be combined in order to obtain an efficient quantum

2 Advances in Mathematical Physics

technology. This calls for qubits that are good for storage such as atoms to be used at quantumnetworks nodes while qubits that have desirable properties for travel such as photons aswell as a coherence quantum interface between these qubits allowing for the exchange ofinformation .

To the end of realizing an efficient quantum computing architecture, this promisingcomposite qubit approach to a quantum technology has been proposed for ion trap [1]and also for neutral atoms [2]. We on the other hand have proposed a similar approach inconnection with semiconductor-based artificial atoms or quantum dots (QDs) [3]. What doesour scheme consist of? It consists of engineering a photonic crystal chip hosting a quantumnetwork made of QDs spin embedded in defect cavities (storage qubits), which constitutethe nodes of the network. These storage qubits can interact with other storage qubits at otherlocations or nodes by means of single photons (traveling qubits), which are guided throughwaveguides. Interestingly, this coherent interface, which is responsible for the state of thestorage qubits to be mapped onto the traveling qubits or the entanglement between them, isitself a qubit system, the cavity-QED qubit (exchange or interaction qubit). Figure 1 depicts astorage qubit (an electron spin in a QD) interacting with a traveling qubit (a single photon)inside a quantum network node.

Our approach to a quantum network offers unique benefits with respect to the othercomposite qubit schemes. For instance, realistic on-chip implementation using photoniccrystal has been shown to be plausible [4]; which is not the case in both the ion trap andneutral atom qubits. In addition, even though low temperatures are desirable for minimizingdecoherence for the QD spin qubit, it is no where near the extreme temperatures needed forthe functioning of the superconducting qubit. They are also much more robust against theinfluence of temperature than ion trap qubits. In fact, spin lifetimes up to 20 millisecondshave been reported [5]. This technology is easily scalable as additional nodes for thequantum network are generated by just creating additional cavities with embedded QDs inthe photonic chip. Besides, because it is a semiconductor-based quantum technology, it isanticipated to be CMOS compatible and cost effective. The use of single photons as travelingqubits as well as the wavelengths considered makes not only the on-chip transfer of quantuminformation but also the long distance quantum communication by means of optical fibersefficient.

In these many regards, combining subwavelength photonic structures and semicon-ductor quantum dots provides an unmatched environment for the implementation of storage,exchange, and travelling qubits in comparison to other composite qubit schemes. Scaling upsuch existing technology by means of computer models is critical in order to build a fullyfunctional quantum network. Such model is expected to lead to a thorough understanding ofnot only the coherent interaction between a QD and a single photon but also the entanglementdynamics between such particles produced inside quantum networks in a controlled way, inparticular the fidelity or amount of such entanglement. For that reason, computer models area fundamental step in gaining control over quantum information. As a result, we investigatein this paper the dynamics of entanglement between a QD spin qubit and a single-photonqubit inside a quantum network node, as well as its robustness against various decoherenceprocesses.

2. Theory of Quantum Network Nodes

Modeling quantum network nodes requires the ability to describe the following physicalsystems �1� a QD, �2� a single-photon field, and �3� the interaction between these two in

Advances in Mathematical Physics 3

g

γ12

Γcavity

Figure 1: Storage, exchange and traveling qubits inside a quantum network node.

a nanocavity. How though are descriptions of the physical systems making up quantumnetwork nodes used to implement the various qubits? First, the storage qubit is implementedusing the spin states of a single excess electron in the conduction band of the QD. What aboutthe traveling qubit? The polarization states of the single-photon in the single mode cavity,whether in the linear or circular polarization eigenbasis, could be used to describe the statesof the traveling qubit. It turns out that the implementation of the quantum network nodesusing semiconductor QD will necessitate the representation of the traveling qubit to be in thecircular polarization eigenbasis. Last, the exchange qubit takes on the form of the “dressed”states in the “dressed” atom picture.

2.1. Modified Jaynes-Cummings Model

Quantum network nodes can be effectively described by a model very similar to the Jaynes-Cummings Model. The main difference between the model for the quantum network nodesand the JC model is the construction of the two-level system, which it is 2-fold degenerate inthe excited state and 4-fold degenerate in the ground state, shown in Figure 2. It is importantto note that these four degenerate ground states are not connected specifically with light andheavy holes, although the selection rules are identical. We will refer to these degenerate statesas �h3�2� (2-fold degenerate valence band states with angular momentum projection �3�2)and �h1�2� (2-fold degenerate valence band states with angular momentum projection �1�2)[6, 7]. It is assumed that the QD is spherical in shape resulting in the confining potential withsymmetry approximately identical to that of the first Brillouin zone of the anticipated cubiclattice of the semiconductor crystal. Accordingly, only a shift in the energy levels occurs whilethe degeneracy between �h3�2� and �h1�2� states is conserved (A degeneracy lift would notprevent this scheme from working [8], it would only change the time and condition necessaryto perform entanglement.) When idle, the quantum dot system has all its ground state levelsoccupied while only one electron occupies the 2-fold degenerate excited state resulting ideallyin an “infinite” decay time. The ability for the excited state to retain this electron is critical tostore and process quantum information in our scheme as we will see in the rest of this section.Last, unlike the Jaynes-Cummings Model, the model for the quantum network nodes willtake account of various decoherence processes.

Because the QD system as a pseudo two-level system can be expressed in the totalangular momentum (orbital angular momentum and spin) eigenbasis, there are clearly


Ec

Ev

∣∣∣∣12,± 1

2

⟩

e

∣∣∣∣

32,± 3

2

⟩

h3/2

∣∣∣∣

32,± 1

2

⟩

h1/2

Figure 2: Two-level approximation of QD for the quantum network nodes.

defined optical transition rules for the electrical dipole interaction between ground andexcited states, namely, that the orbital angular momentum quantum number l changes by �1,and the spin is conserved as well as the parity of the envelop function. Furthermore, it followsthat in the Faraday geometry, which requires the quantization axis of the excess electron spinto be parallel to the direction of light propagation, the empty state in the conduction bandis only populated by circular polarized light (either �σ�� or �σ�� depending on the excesselectron spin state). This is illustrated in Figure 3 for the case when the excess electron spin isinitialized to � ��.

Consequently, the total Hamiltonian for describing the quantum network node systemis thus written as

�H � �HQD � �HPhoton � �HQD-Photon, (2.1)

where

�HQD � �hω3�2�σ3�2v,3�2v ��hω3�2�σ�3�2v,�3�2v

� �hω1�2�σ1�2v,1�2v ��hω1�2�σ�1�2v,�1�2v

� �hωe�σ1�2c,1�2c ��hωe�σ�1�2c,�1�2c,

�HPhoton ��hν�a†

σ��aσ� � �hν�a†σ��aσ�,

�HQD-Photon ��hg3�2�a†

σ��σ�3�2v,�1�2c � �hg3�2�a†σ��σ3�2v,1�2c

� �hg3�2�σ�1�2c,�3�2v�aσ� � �hg3�2�σ1�2c,3�2v�aσ�� hg1�2�a†

σ��σ�1�2v,1�2c � �hg1�2�a†σ��σ1�2v,�1�2c

� �hg1�2�σ1�2c,�1�2v�aσ� � �hg1�2�σ�1�2c,1�2v�aσ�.

(2.2)

The various coupling strengths from valence band states with total angular momentum j ��3�2 to conduction band states with total angular momentum j � �1�2 are be denoted g3�2,and the various coupling strengths from valence band states with total angular momentumj � �1�2 to conduction band states with total angular momentum j � �1�2 are denoted g1�2.


No exciton J = 3/2 hole exciton J = 1/2 hole exciton

σ+z σ−

z

σ+z σ−

z∣∣∣∣

32,− 3

2

⟩

h3/2

∣∣∣∣

12,− 1

2

⟩

e

∣∣∣∣

32,+

12

⟩

h1/2

∣∣∣∣

12,− 1

2

⟩

e

Figure 3: Dipole selection rules in the quantum network nodes.

2.2. Entanglement Process

From dipole selection rules, it can easily be shown that the coupling strengths associatedwith e�h1�2 and e�h3�2 excitons can be expressed as g1�2 �

��1�3�g3�2. This unbalance in thetransition strengths between the e�h1�2 and e�h3�2 excitons is what makes possible both theprocess of mapping out quantum information from the storage qubit onto the traveling qubitand creating entanglement between them. How? Because linearly polarized light is nothingbut a balanced superposition of right- and left-hand circular polarized light such that �ψp� �� σ�z � � �σ�z ��2, the right and left circular components of the polarization accumulatedifferent phases due to the unbalance in transition strengths resulting in the rotation of thesingle-photon linear polarization. This is referred to as the single-photon Faraday Effect [9].

As a result, if the spin of the excess electron is initialized to � ��, then the single-photonpolarization rotates in a right-hand circular motion since the right circular polarizationcomponent accumulates a larger phase while interacting with �h3�2� valence states as depictedin Figure 3. However, if the spin of the excess electron is initialized to � ��, then the singlephoton polarization rotates a left-hand circular motion. This Pauli blocking mechanismresulting in the conditional rotation of the single photon linear polarization based on thestate of the excess electron spin can effectively be used to encode or map quantum statefrom the storage qubit onto the traveling qubit or even for creating entanglement betweenthem.

2.3. Creating Entanglement

Quantum entanglement is a well-established quantum property that has no counterpart inclassical physics; it occurs when the total wave function of the mixed system cannot bewritten in any basis, as a direct product of independent substates (i.e., tensor product). Asa result, the system of the two entangled qubits individually represented, for instance, bystates �0� and �1� has four new computational basis states designated �00�, �01�, �10�, and �11�.In the quantum network node, in order to create entanglement, the spin state of a single excesselectron in the conduction band of the QD must be initialized to �ψe� � �1�2�� ; thisis depicted in Figure 4.


σ−z σ+

z

σ+z σ

+z

|Ψs

⟩=

1√2(| ↑〉 + | ↓ ⟩)

|Ψp

⟩= | �〉 |Ψp

⟩=

1√2(| ⟩

+ | ⟩)

Figure 4: Spin state initialization for entanglement.

Under these conditions, it is unclear in which direction does the polarization of thesingle photon rotates. The entanglement between the QD excess electron spin and the singlephoton polarization is predicted to be the greatest at what would correspond to a 45-degreerotation of the linear polarization of the single photon or ϕ � π�4 [3]. Transforming to a Bellstate eigenbasis for the storage qubit (electron spin) and traveling qubit (photon), we thushave the following maximally entangled Bell state:

ψep� � eish3�2o ψX3�2� � eish1�2o ψX1�2

�� eis

h3�2o

12��σ�� σ�� eish1�2o

12��σ�� σ��

� e�iθ�� e�iϕ � �� eiϕ

2.

(2.3)

3. Modeling Quantum Network Nodes

The density matrix formalism [10, 11], which is an elegant formulation of quantummechanics, is used for the modeling of our quantum network. Most importantly, using thedensity matrix formalism to describe a composite quantum system such as quantum networknodes is indispensable for the analysis of quantum entanglement. This is because the densitymatrix is able to describe correlations between observables in the various subsystems unlikethe Schrodinger’s equation formalism. For instance, the entanglement of two qubits forminga composite system represented by the density matrix �ρ can be computed directly as eitherthe Von Neumann entropy [12] or the normalized linear entropy [13]. So in order to study


Z

X Y

t = 0

t = 0

t = 0

t = 0

t = 0

t = 0

(a)

Z

Xt = 0t = 0t = 0

t = 0

t = 0

t = 0

(b)

Z

Y

t = 0

t = 0

t = 0

t = 0

t = 0

t = 0

α = 0.5, β = 0.5α = 0.4, β = 0.6α = 0.3, β = 0.7

α = 0.2, β = 0.8α = 0.1, β = 0.9α = 0, β = 1

(c)

Y

Xt = 0 t = 0

α = 0.5, β = 0.5α = 0.4, β = 0.6α = 0.3, β = 0.7

α = 0.2, β = 0.8α = 0.1, β = 0.9α = 0, β = 1

(d)

Figure 5: Excess electron spin dynamics on the bloch sphere for small Δ.

the dynamics of the entanglement between qubits inside a quantum network node, all thatis needed is the time evolution of the density matrix describing the subsystems making upthe quantum network node, which is obtained by setting up properly an equation of motionand then solving for it. This is often referred to as the Louiville or Von Neumann Equation ofmotion for the density matrix, and it is shown in

d

dt�ρ � �

i�h ��H,�ρ� � 12��Γ,�ρ�. (3.1)


σ+

σ−

−45◦

45◦�

↔

t = 0

(a)

σ+

σ−

� ↔t = 0

(b)

σ+

σ−

−45◦ 45◦t = 0

α = 0.5, β = 0.5α = 0.4, β = 0.6α = 0.3, β = 0.7

α = 0.2, β = 0.8α = 0.1, β = 0.9α = 0, β = 1

(c)

�

↔

−45◦

45◦

t = 0

α = 0.5, β = 0.5α = 0.4, β = 0.6α = 0.3, β = 0.7

α = 0.2, β = 0.8α = 0.1, β = 0.9α = 0, β = 1

(d)

Figure 6: Photon polarization dynamics on the poincare sphere for small Δ.

3.1. Master Equation

The Louiville or Von Neumann Equation is the most general form of a master equation for asystem whose states are described in term of a density matrix �ρ and whose interactions aredescribed according to the Hamiltonian matrix �H. However, because relaxation processesare more complicated than just the anticommutator of a single relaxation matrix �Γ with


the density matrix �ρ, we use the following master equation, which is more suitable for themodeling of our quantum network using two relaxation matrices �W and �γ ,

dρmm�

dt� �

i�h�k �Hmkρkm� � ρmkHkm�� δmm� �k �m

ρkkWmk � γmm�ρmm� , (3.2)

where Wmk are transitions rates affecting the diagonal elements of the density matrix andγmm� are decoherence rates affecting both diagonal and off-diagonal elements of the densitymatrix. The matrix �ρ is an N �N matrix with m � 1,2, . . . ,N and m�

� 1,2, . . . ,N.

3.2. Matrix Transformations

Our problem is currently expressed in terms of the subsequent matrix equation

��ρ�� H��ρ� � ��ρ��H� � δ��ρ�� W� � ��γ��ρ�, (3.3)

where δ��ρ�� W� � δmm��k �m ρkkWmk, and ��γ��ρ� � ��γ � �ρ�mm� � γmm� � ρmm� . However, theRunge-Kutta algorithm we are using to solve this system of 1st-order Ordinary DifferentialEquations numerically requires that the problem is expressed in the following form:

�y�� y�. (3.4)

Equation (3.3) must be rewritten in the form of (3.4). This means rearranging thedensity matrix �ρ into a column vector y, and the commutator ��H, �� W � �γ into a matrix�� such that

��ρ� TL�� y�1�,�mm��, (3.5)

��H, ��mm�

� �W ��γ� TL�� mm��,�nn��. (3.6)

These operations amount to a transformation to Louiville space where �L�mm��,�nn��, is asuperoperator (tetradic matrices) acting on that space with dimension N2. Equation (3.6)can be broken down into the following three matrix transformations

��H, ��mm�

� TL�� mm��,�nn�� , (3.7)

� �Wmm�� TL�� Γ1�mm��,�nn��, (3.8)

��γmm�� TL�� Γ2�mm��,�nn��, (3.9)

such that

��mm��,�nn�� mm��,�nn��

�Γ1�mm��,�nn�� Γ2�mm��,�nn��. (3.10)


3.3. Algorithms

The first operation transforming �ρmm� to y�1�,�mm�� is quite straightforward for an arbitrarysize square matrix. The essence of the algorithm is to create a column vector y whose numberof elements is N �N and then fill it up using one row of the density matrix at the time.

The second operation transforming an arbitrary size square matrix ��H, ��mm� to��mm��,�nn�� is more complex. Before generating algorithms for problems of arbitrary size, itis useful to work out a simple example. Let assume that the density �ρ and Hamiltonian �Hmatrices are 2�2 matrices so that m � 1,2 and m�

� 1,2 and therefore N � 2

�ρ �

��ρ11 ρ12

ρ21 ρ22

��, �H �

��H11 H12

H21 H22

��. (3.11)

The commutator of the Hamiltonian and the density matrix in matrix form is

��H,�ρ� �

��

�H12ρ21 �H21ρ12��H11ρ12 �H12ρ22

�H12ρ11 �H22ρ12

��H21ρ11 �H22ρ21

�H11ρ21 �H21ρ22

��H21ρ12 �H12ρ21�

��

. (3.12)

Next, the flowing operation ��H, ��2�2 � � ��4�4 is performed on the Hamiltonian �Hsuch that �d�dt��ρ�mm� � ��i��h��H,�ρ�mm� �� d�dt�y�1�,�mm��

��mm��,�nn��y�1�,�mm��,

�H��H11 H12

H21 H22

��

��

0 �H21 H12 0

�H12 H11 �H22 0 H12

H21 0 H22 �H11 �H21

0 H21 �H12 0

��

. (3.13)

The algorithm for an arbitrary square matrix �� of size N2 by N2 is considered next. First, weneed to define new indices i and j which are used to identify sub-block matrices. So (3.13)becomes

�H��H11 H12

H21 H22

��

��

��0 �H21

�H12 H11 �H22

��i�1,j�1��

H12

H12

��i�1,j�2��H21

H21

��i�2,j�1��H22 �H11 �H21

�H12 0

��i�2,j�2

��

. (3.14)


The strategy adopted to fill this matrix up consists of dividing the matrix �� intosmaller quadrants or matrices �Lij of size N �N

��

��

��L11� ��L12� � ��L1m��L21� �

�

��Lm1� ��Lmm��

��

. (3.15)

The off-diagonal matrices �Lij �i� j� identified with the indices i and j are filled with elementfrom the original Hamiltonian ��H, ��mm� using the same indices i and j such that matrices �Lijare themselves diagonal matrices filled with elements Hij of �H. Therefore

�Li� j �

��

Hij 0 0 0

0 Hij 0 0

0 0 � 0

0 0 0 Hij

��

. (3.16)

As for the diagonal matrices ��Lij �i � j��, they are constructed such that

�Li�j �

��

Hij �H1,1 �H1,2 �H1,3 � �H1,m�

�H2,1 Hij �H2,2 �H2,3 � �H2,m�

�H3,1 �H3,2 Hij �H3,3 � �H3,m�

� �

�Hm,1 �Hm,2 �Hm,3 � Hij �Hm,m�

��

. (3.17)

Finally, the relaxation matrices �W and �γ also need to be transformed. In the originalmaster equation, �W and �γ are N �N matrices and are written as

�Wmm� �

��

W11 W12 W13 � W1m�

W21 W22 W23 � W2m�

W31 W32 W33 �

� � W�m�1�m�

Wm1 Wm2 � Wm�m��1� Wmm�

��

,

�γmm� �

��

γ11 γ12 γ13 � γ1m�

γ21 γ22 γ23 � γ2m�

γ31 γ32 γ33 �

� � γ�m�1�m�

γm1 γm2 � γm�m��1� γmm�

��

,

(3.18)


First, �Γ1 matrix is considered. It is built both from relaxation matrices �W and �γ in connectionwith the diagonal elements of the density matrix. When m � m�, the term relating to thetransition rates in (3.2) in matrix form is written as

δ��ρ�� W� � δmm� �k �m

ρkkWmk

�

��

�k

ρkkW1k 0 � 0

0 �k

ρkkW2k 0 0

�

0 0 � �k

ρkkWmk

��

,

(3.19)

whereas the term relating to the decoherence rates in (3.2) in matrix form is written

��γ��ρ� � γmmρmm� ρmm�

k

Wkm

�

��

ρ11�k

Wk1 0 � 0

0 ρ22�k

Wk2 0 0

�

0 0 � ρkk�k

Wkm

��

,

(3.20)

since

γmm �12�k

�Wkm �Wkm�� 12�k

�Wkm �Wkm� � �k �m

Wkm. (3.21)

Combining (3.19) and (3.20), we get the following new term where �W and �γ areforming the matrix �W�,

��ρ�� W�� δ��ρ�� W� � ��γ��ρ�� δmm� �

k �m

ρkkWmk � ρmm�k

Wkm

�

��

�k

ρkkW1k � ρ11�k

Wk1 0 � 0

0 �k

ρkkW2k � ρ22�k

Wk2 0 0

�

0 0 � �k

ρkkWmk � ρkk�k

Wkm

��

.

(3.22)


Before coming with an algorithm that can transform �W� to �Γ1 for a problem of arbitrarysize, it is useful to work out a simple example. It is assumed for the moment that �ρ, �W, and �γare 2 � 2 matrices such that m � 1,2 and m�


�ρ �

��ρ11 ρ12

ρ21 ρ22

��, �W �

��W11 W12

W21 W22

��, �γ �

��γ11 γ12

γ21 γ22

��

��W21 0

0 W12

��. (3.23)

From the master equation, the terms related to transition and decoherence rates are thereforeexpressed as

δ��ρ�� W� � ��γ��ρ� � δmm� �k �m

ρkkWmk � ρmm�k

Wkm

�

��ρ22W12 � ρ11W21 0

0 ρ11W21 � ρ22W12

��.

(3.24)

In Louiville space, the term δ��ρ�� W� � ��γ��ρ� is rewritten yT�1�,�4�

�Γ1�4�,�4� such that

yT�1, J� �Γ1�I, J�

�ρ11 0 0 ρ22�

��

�W21 0 0 W21

0 0 0 0

0 0 0 0

W12 0 0 �W12

��

.(3.25)

Thus, in order to obtain �Γ1 for an arbitrary size problem, the strategy is to create a squarematrix of size N2

�N2 such that the term δ��ρ�� W��γ��ρ� is rewritten yT�1�,�mm��

�Γ1�mm��,�nn��,

and then fill �Γ1. The notation for matrix indices in state space is as follows i � 1,2, . . . ,N andj � 1,2, . . . ,N. Their counterparts in Louiville space are I � 1,2, . . . ,N2 and J � 1,2, . . . ,N2.The key is to identify the columns (or J ’s) in �Γ1 that needed to be filled, and then fill out theappropriate rows (or I’s) with the appropriate rates. The columns or J ’s to be filled are theones whose indices are matching the J ’s in yT�1, J� that correspond to diagonal terms in theoriginal density matrix �ρmm� , namely, ρm�1,m��1 � y1,J�1 and ρm�2,m��2 � y1,J�4 as shown inthe example above. A general expression for the columns to be filled is

J � �j � 1� �N � j, where j � 1,2, . . . ,N. (3.26)

It turns out that the set of indices for rows and columns to be filled are identical (I � J);therefore determining the indices for the columns to be filled also give the ones for the rows.

Therefore, the algorithm to generate elements of �Γ1 for a problem of an arbitrary sizeis as follows

�Γ1�I, J� ��

�W�i, j� �for I � J�,��γ�i, j� �for I � J�, (3.27)


where I � �i�1� �N � i and J � �j �1� �N � j with i � 1,2, . . . ,N and j � 1,2, . . . ,N. In Louivillespace, when considering only the diagonal terms of the density matrix, �Γ1 can be written as

�W �

��

��k � 1

Wk1 W21 W31 � Wk1

W12 ��k � 2

Wk2 W32 � Wk2

W13 W23 ��k � 3

Wk3 �

� � Wk�m�1�

W1m W2m � W�k�1�m ��k �m

Wkm

��

. (3.28)

Next, �Γ2 is considered. Unlike �Γ1, �Γ2 is associated with the case when m�m� in themaster equation see (3.3). This means that δ��ρ�� W� � 0; therefore, only ��γ��ρ� contributesto the master equation. Furthermore, �Γ2 is generated from only the decoherence rates γmm� ,which are off-diagonal elements of �γ such that

�γm�m� �12�k

�Wmk �Wkm��. (3.29)

These rates are included in the master equation by means of a Hadamard or Schur productbetween the relaxation matrix �γ and the density matrix �ρ such that

��γ��ρ� � γmm� � ρmm�

�

��

0 γ12ρ12 γ13ρ13 � γ1m�ρ1m�

γ21ρ21 0 γ23ρ23 � γ2m�ρ2m�

γ31ρ31 γ32ρ32 0 �

� � γ�m�1�m�ρ�m�1�m�

γm1ρm1 γm2ρm2 � γm�m��1�ρm�m��1� 0

��

.(3.30)

Before coming with an algorithm that can transform �γ to �Γ1 for a problem of arbitrarysize, it is useful to work out a simple example. It is assumed for the moment that �ρ and �γ are2 � 2 matrices such that m � 1,2 and m�


�ρ �

��ρ11 ρ12

ρ21 ρ22

��,

�γ i� j ��

0 γ12

γ21 0

��

��0 W12

W21 0

��.

(3.31)


From the master equation, the terms related to decoherence rates are thereforeexpressed as

��γ��ρ� � γmm� � ρmm� �

��0 ρ12W12

ρ21W21 0

��. (3.32)

In Louiville space, the term ��γ��ρ� is rewritten �Γ2�4�,�4�y�1�,�4� such that

�Γ2�I, J�y γ�I,1��

0 0 0 0

0 γ12 0 0

0 0 γ21 0

0 0 0 0

��

��

ρ11

ρ12

ρ21

ρ22

��

.(3.33)

Consequently, in order to obtain �Γ2 for an arbitrary size problem, the strategy is to createfirst a vector γ�mm��,�1� from �γmm� whose number of elements is N2 and then fill it up byconcatenating the rows of the density matrix at the time. Next, �Γ2, a square matrix of sizeN2

�N2, is generated by filling its diagonal elements with elements from the vector γ�mm��,�1�.Evidently, the matrix �Γ2 always consists of a diagonal matrix. The master equation in (3.3)now takes on the following form:

�y�� y� � �y�T��Γ1� � ��Γ2��y�. (3.34)

3.4. Solving for Entanglement

The entanglement of two qubits forming a composite system represented by the densitymatrix ρA�B can be computed directly as either the Von Neumann entropy [12] or thenormalized linear entropy [13] of the reduced density matrix of either of the two qubits.Equations (3.35) below show, respectively, the Von Neumann and normalized linear entropies

SVN � �TrA�TrBρA�Blog2�TrBρ

A�B �,SNL � 2�1 � TrA�TrBρ

A�B 2�.(3.35)

Whether it is calculated from the Von Neumann entropy or the normalized linear entropy, theentanglement between two qubits ranges from 0 to 1, where 1 means that they are maximallyentangled.


Let us consider the “pure” state density matrix describing the combined spin-photonsystem within a single high-Q cavity shown in (3.36). It contains time-dependent variablesρij , which describe the probability of occupying each distinct set of states in the cavity system,which are � �,σ�z �, corresponding to the state when the excess electron spin being down witha left circularly polarized (LCP or σ�z ) photon in the cavity, � �,σ�z � to the excess electron spinbeing down with a right circularly polarized (RCP or σ�z ) photon in the cavity, � �,X1�2� to theexcess electron spin being down with a e � h1�2 exciton (trion), � �,X3�2� to the excess electronspin being down with a e � h3�2 exciton (trion), � �,σ�z � to the excess electron spin being upwith an LCP photon in the cavity, � �,σ�z � to the excess electron spin being up with a RCPphoton in the cavity, � �,X1�2�, to the excess electron spin being up with a e � h1�2 exciton(trion), and � �,X3�2� to the excess electron spin being up with a e � h3�2 exciton (trion)

�ρ � �ψ�t�� ψ�t��

�

�,σ�z � �,σ�z � !�,X1�2 !�,X3�2 �,σ�z � �,σ�z � !�,X1�2 !�,X3�2��

ρ11

ρ21

ρ31

ρ41

ρ51

ρ61

ρ71

ρ81

ρ12

ρ22

ρ32

ρ42

ρ52

ρ62

ρ72

ρ82

ρ13

ρ23

ρ33

ρ43

ρ53

ρ63

ρ73

ρ83

ρ14

ρ24

ρ34

ρ44

ρ54

ρ64

ρ74

ρ84

ρ15

ρ25

ρ35

ρ45

ρ55

ρ65

ρ75

ρ85

ρ16

ρ26

ρ36

ρ46

ρ56

ρ66

ρ76

ρ86

ρ17

ρ27

ρ37

ρ47

ρ57

ρ67

ρ77

ρ87

ρ18

ρ28

ρ38

ρ48

ρ58

ρ68

ρ78

ρ88

��

��,σ�z ��,σ�z ��,X1�2��,X3�2��,σ�z ��,σ�z ��,X1�2��,X3�2�

(3.36)

Next, in order to solve for the entanglement dynamics inside our quantum network node, weneed first the Hamiltonian, which is obtained in the rotating frame, as well as the relaxationmatrices �W and �γ

�H �

�,σ�z � �,σ�z � !�,X1�2 !�,X3�2 �,σ�z � �,σ�z � !�,X1�2 !�,X3�2��

�hν0

0

Vh3�2

0

0

0

0

0�hνVh1�2

0

0

0

0

0

0

Vh1�2

Ec

0

0

0

0

0

Vh3�2

0

0

Ec

0

0

0

0

0

0

0

0�hν0

Vh1�2

0

0

0

0

0

0�hν0

Vh3�2

0

0

0

0

Vh1�2

0

Ec

0

0

0

0

0

0

Vh3�2

0

Ec

��

��,σ�z ��,σ�z ��,X1�2��,X3�2��,σ�z ��,σ�z ��,X1�2��,X3�2�

(3.37)


where Ec is the energy of the excess electron, ν the photon frequency, Vh3�2�

�hg3�2 theinteraction energy associated with the e�h3�2 excitons, and Vh1�2

��hg1�2 the interaction energy

associated with the e � h1�2 excitons

"W�

�

�,σ�z � �,σ�z � !�,X1�2 !�,X3�2 �,σ�z � �,σ�z � !�,X1�2 !�,X3�2��

�W15

000

W15

000

0�W26

000

W26

00

0W23

�W23�W34�W37�W38

W34

00

W37

W38

W14

0W34

�W14�W34�W47�W48

00W47

W48

W15

000

�W15

000

0W26

000

�W26

00

00W37

W47

W57

0�W57�W37�W47�W78

W78

00

W38

W48

0W68

W78

�W68�W38�W48�W78

��

��,σ�z ��,σ�z ��,X1�2��,X3�2��,σ�z ��,σ�z ��,X1�2��,X3�2�

(3.38)

where W23, W14, W57, and W68 are nonreversible decay rates, respectively, from the trion state� �,X1�2� to the photonic state �,σ�z �, from the trion state � �,X3�2� to the photonic state �,σ�z �,from the trion state � �,X1�2� to the photonic state �,σ�z �, and from the trion state � �,X3�2� tothe photonic state �,σ�z �. Next, the Fermi contact terms are depicted by W15 representing atransition for the excess electron spin from � �,σ�z � to �,σ�z � and W26 representing a transitionfor the excess electron spin from � �,σ�z � to �,σ�z �. Last, the hole hyperfine interaction terms aredepicted by W37, W48, W34, W78, W38, and W47. They, respectively, represent the transitionfrom the following trion state � �,X1�2� to the trion state �,X1�2�, the transition from thefollowing trion state � �,X3�2� to the trion state �,X3�2�, the transition from the following trionstate � �,X1�2� to the trion state � �,X3�2�, the transition from the following trion state � �,X1�2� tothe trion state �,X3�2�, the transition from the following trion state � �,X1�2� to the trion state �,X3�2�, and the transition from the following trion state � �,X3�2� to the trion state � �,X1�2�

�,σ�z � �,σ�z � !�,X1�2 !�,X3�2 �,σ�z � �,σ�z � !�,X1�2 !�,X3�2

�γ �

��

0

γ21

γ31

γ41

γ51

γ61

γ71

γ81

γ12

0

γ32

γ42

γ52

γ62

γ72

γ82

γ13

γ23

0

γ43

γ53

γ63

γ73

γ83

γ14

γ24

γ34

0

γ54

γ64

γ74

γ84

γ15

γ25

γ35

γ45

0

γ65

γ75

γ85

γ16

γ26

γ36

γ46

γ56

0

γ76

γ86

γ17

γ27

γ37

γ47

γ57

γ67

0

γ87

γ18

γ28

γ38

γ48

γ58

γ68

γ78

0

��

��,σ�z ��,σ�z ��,X1�2��,X3�2��,σ�z ��,σ�z ��,X1�2��,X3�2�

(3.39)

where each element γij is calculated according to (3.29) .


Then, the amount of entanglement between the excess electron spin and the singlephoton polarization can be worked out from the density matrix. The normalized linearentropy is considered

�ρ � �ψ�t�� ψ�t��

��,σ�z � ��,σ�z � ��,X1�2� ��,X3�2� ��,σ�z � ��,σ�z � ��,X1�2� ��,X3�2��

��

�ρ��σσ

��

�ρ��

σX

��

�ρ��σσ

��

�ρ��

σX

�

��

�ρ��

Xσ

��

�ρ��

XX

��

�ρ��

Xσ

��

�ρ��

XX

�

��

�ρ��σσ

��

�ρ��

σX

��

�ρ��σσ

��

�ρ��

σX

�

��

�ρ��

Xσ

��

�ρ��

XX

��

�ρ��

Xσ

��

�ρ��

XX

�

�

��,σ�z �

��,σ�z �

��,X1�2

��,X3�2

��,σ�z �

��,σ�z �

��,X1�2

��,X3�2

(3.40)

Tracing the density matrix over the cavity polariton results in the spin reduced density matrixsuch that

TrPolariton�ρ � �ρred,spin

� TrPhoton�ρ � TrExcitonρ

� σ�z ��ρ�σ�z � � σ�z ��ρ�σ�z � � !X3�2��ρ�X3�2� � !X1�2��ρ�X1�2�

�

��Tr�ρ��

σσ Tr �ρ��

σσ

Tr�ρ��

σσ Tr �ρ��

σσ

��

��Tr�ρ��

XX Tr �ρ��

XX

Tr�ρ��

XX Tr �ρ��

XX

��

��Tr�ρ��

σσ � Tr �ρ��

XX Tr �ρ��


XX

Tr�ρ��


XX Tr �ρ��


XX

�� 2 � 2,

(3.41)

where

�ρ��

σσ �

��ρ��σ�z σ�z ρ��σ�z σ�z

ρ��σ�z σ�zρ��σ�z σ�z

��ρ��

σσ �



��,

�ρ��

σσ �



��ρ��

σσ �



��,

�ρ��

XX �

��ρ��X3�2X3�2

ρ��X3�2X1�2

ρ��X1�2X3�2ρ��X1�2X1�2

��ρ��

XX �

��ρ��X3�2X3�2

ρ��X3�2X1�2

ρ��X1�2X3�2ρ��X1�2X1�2

��,

�ρ��

XX �

��ρ��X3�2X3�2

ρ��X3�2X1�2

ρ��X1�2X3�2ρ��X1�2X1�2

��ρ��

XX �

��ρ��X3�2X3�2

ρ��X3�2X1�2

ρ��X1�2X3�2ρ��X1�2X1�2

��.

(3.42)


Entanglement

×10−101.210.80.60.40.20

Time (s)

α = 0.5, β = 0.5α = 0.4, β = 0.6α = 0.3, β = 0.7

α = 0.2, β = 0.8α = 0.1, β = 0.9α = 0, β = 1

0

0.5

1

1.5

2

2.5

3

Ent

ropy

Figure 7: Spin-photon entanglement dynamics for small Δ.

Next, the 2�2 matrix TrPolariton�ρ needs to be squared. As a result, we obtain

�TrPolariton�ρ�2�

��Tr �ρ��

σσ � Tr�ρ��

XX Tr�ρ��


XX

Tr �ρ��

σσ � Tr�ρ��

XX Tr�ρ��


XX

��

2

�

��

��,

(3.43)

where � denotes#$

Tr�ρ��

σσ

�

Tr�ρ��

XX

%&, � denotes

#$

Tr�ρ��

σσ

�

Tr�ρ��

XX

%&, � denotes

#$

Tr�ρ��

σσ

�

Tr�ρ��

XX

%&, and � denotes

#$

Tr�ρ��

σσ

�

Tr�ρ��

XX

%&.

Finally, we take the trace of this square matrix over the spin such that

η � Trspin�TrPolariton�ρ�2�

��

#'''$

Tr�ρ��

σσ

�

Tr�ρ��

XX

%(((&

2

�

#'''$

Tr�ρ��

σσ

�

Tr�ρ��

XX

%(((&

#'''$

Tr�ρ��

σσ

�

Tr�ρ��

XX

%(((&

��

�

��

#'''$

Tr�ρ��

σσ

�

Tr�ρ��

XX

%(((&

#'''$

Tr�ρ��

σσ

�

Tr�ρ��

XX

%(((&�

#'''$

Tr�ρ��

σσ

�

Tr�ρ��

XX

%(((&

2

��

. (3.44)

The normalized linear entropy is then

SNL � 2�1 � η�. (3.45)


EM field

×10−101.210.80.60.40.20

Time (s)

σ+

α = 0.5, β = 0.5α = 0.4, β = 0.6α = 0.3, β = 0.7α = 0.2, β = 0.8α = 0.1, β = 0.9α = 0, β = 1

σ−

α = 0.5, β = 0.5α = 0.4, β = 0.6α = 0.3, β = 0.7α = 0.2, β = 0.8α = 0.1, β = 0.9α = 0, β = 1

0

0.10.2

0.30.40.5

0.6

0.70.80.9

1

Prob

abili

tyam

plit

udes

Figure 8: Right and left circular polarization prob. amplitudes for small Δ.

4. Results

It is assumed that a GaAs/InGaAs QD with emission wavelength λQD of 1.182μm, whichcorresponds to the frequencyωQD � 1.594�1015 rad�s, is placed inside a photonic crystal cavity.Next, assuming the dipole moment associated with the InGaAs QD for interactions involving�h3�2� electrons is μvc � 29D, the frequency of the photon ω � ωQD � Δ � 1.594 � 1015 rad�s,and the cavity mode volume V0 � 0.039μm3, the interaction frequency g3�2 involving thevalence states �h3�2� was found to be 132 � 109 rad�s leading to the interaction energy in theHamiltonian Vh�3�2�

��hg3�2 and the interaction frequency g1�2 involving the valence states

�h1�2� to be g3�2�3 � 76 � 109 rad�s corresponding to the interaction energy Vh1�2 ��hg1�2.

First, the entanglement dynamics is considered without decoherence, and then the effects ofvarious decoherence processes on the entanglement dynamics are investigated.

4.1. Entanglement Dynamics without Decoherence

There are two regimes of interest when studying the dynamics entanglement inside the cavitywith or without decoherence, namely, the small and large energy detuning regimes withrespect to the QD and the field. In both regimes, we consider interaction times needed torotate the linear polarization of the photon by 90 degrees from one pure state to the otherexpecting the condition of maximum entanglement to occur somewhere in the range duringwhich the photon is in a superposition of its polarization eigenstates.

4.1.1. Case 1: Small Δ, Spin Initialized to α� �� β� ��, Photon Initialized to � ��

The “small” detuning energy is selected to be Δ � 78μeV, which corresponds to the optimizedvalue Δ � 0.86g3�2 so as to have a photon and not an exciton back into the cavity once the


polarization has rotated by 90 degrees or 2ϕ � π . This takes approximately 130 picoseconds.The density matrix is initialized to (4.1), where α and β are the probability amplitudesassociated with spin state � �� and spin state � ��, respectively,

�ρ�t � 0� � 12�

�,σ�z � �,σ�z � !�,X1�2 !�,X3�2 �,σ�z � �,σ�z � !�,X1�2 !�,X3�2��

�α�2�α�20

0

β�α

β�α

0

0

�α�2�α�20

0

β�α

β�α

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

α�β

α�β

0

0

�β�2�β�20

0

α�β

α�β

0

0

�β�2�β�20

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

��

��,σ�z ��,σ�z ��,X1�2��,X3�2��,σ�z ��,σ�z ��,X1�2��,X3�2�

(4.1)

The results are shown in Figures 5, 6, 7, and 8 . In the small Δ regime, there are threepeaks or maxima in the calculated electron-photon system entropy, which means that thereare three different conditions for maximum entanglement. This differs somehow from theprediction obtained from the standard parameterization approach [14]. These peaks occurafter 71,93, and 116 picoseconds of interaction time. However, there is a trade-off since theprobability of creating an exciton is nearly 0.4 or 40% for the first 2 peaks and 45% for the lastpeak. This is not desirable from an engineering point of view since the aim would be to havea high probability of releasing the photon back into the quantum network waveguides oncemaximally entangled. Entanglement is also confirmed by the fact that the Poincare sphereshows intermediate polarization states inside the unit sphere .

4.1.2. Case 2: Large Δ, Spin Initialized to α� �� β� ��, Photon Initialized to � ��

Here, the “large” detuning energy is selected to be Δ � 1.5 meV, and the 90 degrees rotationof the linear polarization takes place in approximately 1.225 nanoseconds. The results areshown in Figures 9, 10, 11. and 12 . In the large Δ regime, there is only one peak or maximaof magnitude 1 in the calculated electron-photon system entropy, as predicted in by thestandard parameterization approach [14]. This peak occurs at 0.625 nanoseconds for all cases.Furthermore, the probability of creating an exciton is effectively always zero. This is greatfrom an engineering point of view since there will always be a high probability of releasing aphoton back into the quantum network waveguides once maximally entangled.

4.2. Entanglement Dynamics with Decoherence

Recently, the hyperfine coupling with the nuclear spins of the semiconductor host material(� 105 nuclei in a single quantum dot), which acts as a random magnetic field (up to a few


Z

X Y

t = 0

t = 0

t = 0

t = 0

t = 0

t = 0

(a)

Z

X

t = 0

t = 0

t = 0

t = 0t = 0t = 0

(b)

Z

Y

t = 0

t = 0

t = 0

t = 0

t = 0

t = 0

α = 0.5, β = 0.5α = 0.4, β = 0.6α = 0.3, β = 0.7

α = 0.2, β = 0.8α = 0.1, β = 0.9α = 0, β = 1

(c)

Y

Xt = 0 t = 0

α = 0.5, β = 0.5α = 0.4, β = 0.6α = 0.3, β = 0.7

α = 0.2, β = 0.8α = 0.1, β = 0.9α = 0, β = 1

(d)

Figure 9: Excess electron spin dynamics on the bloch sphere for large Δ.

Tesla in InAs quantum dots), has been identified as the ultimate limit, at low temperature,to the electron spin relaxation or decoherence in quantum dots. Consequently, assuming lowtemperature conditions, only the various decoherence processes associated with hyperfineinteractions will be considered; where as, those associated with phonon interactions areignored.

First, the Fermi contact term is considered. It relates to the direct interaction of thenuclear dipole with the spin dipoles and is only non-zero for states with a finite electron spindensity at the position of the nucleus (those with unpaired electrons in the conduction band).Therefore, the Fermi contact hyperfine interaction does not affect the trion states, just the


σ+

σ−

−45◦

45◦�

↔

t = 0

(a)

σ+

σ−

� ↔t = 0

(b)

σ+

σ−

−45◦ 45◦t = 0

α = 0.5, β = 0.5α = 0.4, β = 0.6α = 0.3, β = 0.7

α = 0.2, β = 0.8α = 0.1, β = 0.9α = 0, β = 1

(c)

�

↔

−45◦

45◦

t = 0

α = 0.5, β = 0.5α = 0.4, β = 0.6α = 0.3, β = 0.7

α = 0.2, β = 0.8α = 0.1, β = 0.9α = 0, β = 1

(d)

Figure 10: Photon polarization dynamics on the poincare sphere for large Δ.

following states � �,σ�z �, � �,σ�z �, � �,σ�z �, and � �,σ�z �. A typical longitudinal relaxation time forsemiconductors is on the order of T1 � 1 ns [15].

Second, due to the p symmetry of the Bloch wavefunctions in the valence band, thecoupling of the nuclei with holes can be generally neglected because the Fermi contactinteraction vanishes. The coupling constants of the hole-nuclear interaction are significantlysmaller than the coupling constants of the electron-nuclear interaction. One recent referencework determined the longitudinal relaxation time for hole-nuclear interaction to be


Entanglement

×10−91.210.80.60.40.20

Time (s)

α = 0.5, β = 0.5α = 0.4, β = 0.6α = 0.3, β = 0.7

α = 0.2, β = 0.8α = 0.1, β = 0.9α = 0, β = 1

0

0.5

1

1.5

2

2.5

3

Ent

ropy

Figure 11: Spin-photon entanglement dynamics for large Δ.

EM field

×10−91.210.80.60.40.20

Time (s)

σ+

α = 0.5, β = 0.5α = 0.4, β = 0.6α = 0.3, β = 0.7α = 0.2, β = 0.8α = 0.1, β = 0.9α = 0, β = 1

σ−

α = 0.5, β = 0.5α = 0.4, β = 0.6α = 0.3, β = 0.7α = 0.2, β = 0.8α = 0.1, β = 0.9α = 0, β = 1

0

0.10.2

0.30.4

0.50.60.70.80.9

1

Prob

abili

tyam

plit

udes

Figure 12: Right and left circular polarization prob. amplitudes for large Δ.

T1 � 14 ns [16]. Furthermore, all trion states are affected, namely, � �,X1�2�, � �,X3�2�, � �,X1�2�,and � �,X3�2�. It is assumed that T1 is the same for both �h1�2�-nucleus and �h3�2�-nucleusinteractions. It is also interesting to note that there are not any dark states for trions.

Last, radiative recombination rates of trions are also included in our model ofdecoherence. In the case of InGaAs quantum dots with an emission wavelength of λ � 950 nm,a radiative recombination lifetime of τX� � 0.84 ns for negatively charged excitons wasreported [17] as well as τX� � 1.24 ns in the case of InAs quantum dots with an emissionwavelength of λ � 850 nm [18]. On the other hand, GaAs quantum dots spin relaxation as


Z

X Y

(a)

Z

X

(b)

Z

Y

α = 0.5, β = 0.5α = 0.4, β = 0.6α = 0.3, β = 0.7

α = 0.2, β = 0.8α = 0.1, β = 0.9α = 0, β = 1

(c)

Y

X

α = 0.5, β = 0.5α = 0.4, β = 0.6α = 0.3, β = 0.7

α = 0.2, β = 0.8α = 0.1, β = 0.9α = 0, β = 1

(d)

Figure 13: Excess electron spin dynamics on the bloch sphere for small Δ.

phonon-assisted Dresselhaus spin-orbit scattering was estimated T1 � 34μs at T � 4.5 K.This illustrates the fact that these types of decoherence processes can be ignored at lowtemperature.

4.2.1. Case 1: Small Δ, Spin Initialized to α� �� β� ��, Photon Initialized to � ��

The “small” detuning energy is again selected to be Δ � 78μeV. The results are shown infigures 13, 14, 15, and 16. In the small Δ regime, for all case of the excess electron spin beingin a superposition of its eigenstates, a decent amount of entanglement (35% entanglement) is

26 Advances in Mathematical PhysicsRCP

LCP

−45

45X

Y

(a)

RCP

LCP

Y X

(b)

RCP

LCP

−45 45

α = 0.5, β = 0.5α = 0.4, β = 0.6α = 0.3, β = 0.7

α = 0.2, β = 0.8α = 0.1, β = 0.9α = 0, β = 1

(c)

Y

X

−45

45

α = 0.5, β = 0.5α = 0.4, β = 0.6α = 0.3, β = 0.7

α = 0.2, β = 0.8α = 0.1, β = 0.9α = 0, β = 1

(d)

Figure 14: Photon polarization dynamics on the poincare sphere for small Δ.

obtained within only the first 200 picoseconds. Afterward, all entanglement is lost. Moreover,the dynamics of entanglement is interesting in that for the lower amounts of superposition inthe excess spin state, maximum entanglement is reached faster, yet it decreases also faster.

4.2.2. Case 2: Large Δ, Spin Initialized to α� �� β� ��, Photon Initialized to � ��

Also, the “large” detuning energy is selected to be Δ � 1.5 meV. A smaller amount ofentanglement is realized, namely, 25%. And, it lasts only within the first 300 picoseconds.


Entanglement

×10−1032.521.510.50

Time (s)

α = 1, β = 0α = 0.9, β = 0.1α = 0.8, β = 0.2

α = 0.7, β = 0.3α = 0.6, β = 0.4α = 0.5, β = 0.5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Ent

ropy

Figure 15: Spin-photon entanglement dynamics for small Δ.

EM field

×10−1032.521.510.50

Time (s)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Prob

abili

tyam

plit

udes

Figure 16: Right and left circular polarization prob. amplitudes for small Δ.

These results are shown in Figures 17, 18, 19, and 20. Here again the dynamics ofentanglement is unexpected as lower amounts of superposition in the excess electron spinstates result in a larger maximum for the entanglement.

5. Conclusion

Some entanglement between an electron spin qubit within a quantum dot and a single-photon qubit interacting inside a high-Q nanocavity can be obtained even in the presence


Z

X Y

(a)

Z

X

(b)

Z

Y

α = 0.5, β = 0.5α = 0.4, β = 0.6α = 0.3, β = 0.7

α = 0.2, β = 0.8α = 0.1, β = 0.9α = 0, β = 1

(c)

Y

X

α = 0.5, β = 0.5α = 0.4, β = 0.6α = 0.3, β = 0.7

α = 0.2, β = 0.8α = 0.1, β = 0.9α = 0, β = 1

(d)

Figure 17: Excess electron spin dynamics on the bloch sphere for large Δ.

of severe decoherence processes. Whether or not such amount of entanglement for such shortperiods of time can be useful in a real physical system remains to be seen. For a certainty, theperformance of such scheme will have to be improved. There are three areas one could lookinto.

First, performances could be improved by increasing interaction frequencies. This isdone by making it a resonant process, or by reducing the volume of the electromagneticmode, or by changing material system so as to obtain a strong dipole moment for thequantum dot or by increasing dramatically the quality factor of the cavity.

Advances in Mathematical Physics 29RCP

LCP

−45

45X

Y

(a)

RCP

LCP

Y X

(b)

RCP

LCP

−45 45

α = 0.5, β = 0.5α = 0.4, β = 0.6α = 0.3, β = 0.7

α = 0.2, β = 0.8α = 0.1, β = 0.9α = 0, β = 1

(c)

Y

X

−45

45

α = 0.5, β = 0.5α = 0.4, β = 0.6α = 0.3, β = 0.7

α = 0.2, β = 0.8α = 0.1, β = 0.9α = 0, β = 1

(d)

Figure 18: Photon polarization dynamics on the poincare sphere for large Δ.

Second, performances could be increased by eliminating or reducing the effect ofdecoherence. Indium atoms has I9/2 spin and arsenic atom have I3/2 spin; which make InAsquantum dots bad candidate as far as decoherence is concern. Other semiconductors withlower or without nucleus spin could be used. The dephasing times of the II–VI compoundsare 3–10 times larger than dephasing times for III–V compounds [19]; however, for wurzite-type semiconductors to have similar optical selection rules as in the case of zinc-blende-typesemiconductors, propagation along the c axis is needed [20].


Entanglement

×10−1032.521.510.50

Time (s)

α = 1, β = 0α = 0.9, β = 0.1α = 0.8, β = 0.2

α = 0.7, β = 0.3α = 0.6, β = 0.4α = 0.5, β = 0.5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Ent

ropy

Figure 19: Spin-photon entanglement dynamics for large Δ.

EM field

×10−1032.521.510.50

Time (s)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Prob

abili

tyam

plit

udes

Figure 20: Right and left circular polarization prob. amplitudes for large Δ.

At last, improvements in performances could also be obtained performing manipula-tions on the nuclear system [21], or using hole spin in the valence band instead of the electronspin in the conduction band as storage qubit since it is less influenced by the nucleus spin.

Acknowledgment

The authors acknowledge the support from NSF ECCS-0725514.


References

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[7] A. I. Ekimov, F. Hache, M. C. Schanneklein, et al., “Absorption and intensity-dependent photolumi-nescence measurements on Cdse quantum dots—assignment of the 1st electronic-transitions,” Journalof the Optical Society of America B, vol. 10, pp. 100–107, 1993.

[8] C. Y. Hu, A. Young, J. L. O’Brien, W. J. Munro, and J. G. Rarity, “Giant optical Faraday rotationinduced by a single-electron spin in a quantum dot: applications to entangling remote spins via asingle photon,” Physical Review B, vol. 78, no. 8, Article ID 085307, 5 pages, 2008.

[9] H. P. Seigneur, M. N. Leuenberger, and W. V. Schoenfeld, “Single-photon Mach-Zehnder interferom-eter for quantum networks based on the single-photon Faraday effect,” Journal of Applied Physics, vol.104, no. 1, Article ID 014307, 2008.

[10] U. Fano, “Description of states in quantum mechanics by density matrix and operator techniques,”Reviews of Modern Physics, vol. 29, pp. 74–93, 1957.

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[13] A. Silberfarb and I. H. Deutsch, “Entanglement generated between a single atom and a laser pulse,”Physical Review A, vol. 69, no. 4, Article ID 042308, 8 pages, 2004.

[14] M. N. Leuenberger, “Fault-tolerant quantum computing with coded spins using the conditionalFaraday rotation in quantum dots,” Physical Review B, vol. 73, no. 7, Article ID 075312, 8 pages, 2006.

[15] I. A. Merkulov, Al. L. Efros, and M. Rosen, “Electron spin relaxation by nuclei in semiconductorquantum dots,” Physical Review B, vol. 65, no. 20, Article ID 205309, 8 pages, 2002.

[16] B. Eble, C. Testelin, P. Desfonds, et al., “Hole-nuclear spin interaction in quantum dots,” PhysicalReview Letters, vol. 102, no. 14, Article ID 146601, 4 pages, 2009.

[17] P. A. Dalgarno, J. M. Smith, J. McFarlane, et al., “Coulomb interactions in single charged self-assembled quantum dots: radiative lifetime and recombination energy,” Physical Review B, vol. 77,no. 24, Article ID 245311, 8 pages, 2008.

[18] G. Munoz-Matutano, J. Gomis, B. Alen, et al., “Exciton, biexciton and trion recombination dynamicsin a single quantum dot under selective optical pumping,” Physica E, vol. 40, no. 6, pp. 2100–2103,2008.

[19] C. Testelin, F. Bernardot, B. Eble, and M. Chamarro, “Hole-spin dephasing time associated withhyperfine interaction in quantum dots,” Physical Review B, vol. 79, no. 19, Article ID 195440, 13 pages,2009.

[20] G. D. Scholes, “Selection rules for probing biexcitons and electron spin transitions in isotropicquantum dot ensembles,” Journal of Chemical Physics, vol. 121, no. 20, pp. 10104–10110, 2004.

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